The present invention relates to methods for controlling the temperature program used with thermogravimetric analyses.
Kinetics is the study of the dependence of a chemical reaction on time and temperature. Kinetic reactions are often described using two equations. The first of these is known as the rate equation and describes the relationship between the rate of reaction, time and amount of material. For homogeneous decomposition or volatilization reactions, the reaction is almost universally found to follow the general rate equation which takes the form:                                           ⅆ            α                                ⅆ            t                          =                                            k              ⁡                              (                T                )                                      ⁡                          [                              f                ⁡                                  (                  α                  )                                            ]                                n                                    (        1        )            
Where:
xcex1=reaction fraction
dxcex1/dt=rate of reaction
k(T)=rate constant at a given temperature T
T=absolute temperature
f(xcex1)=kinetic expression
n=reaction order
The second equation describing kinetic reactions details the dependence of the rate constant on temperature and is known as the Arrhenius equation.
k(T)=Ze(xe2x88x92E/RT)xe2x80x83xe2x80x83(2)
Where:
z=the pre-exponential factor
e=natural logarithm base
E=activation energy
R=gas constant
The rate and Arrhenius equations may be combined into a single form:                                           ⅆ            α                                ⅆ            t                          =                                            Z              ⁡                              [                                  f                  ⁡                                      (                    α                    )                                                  ]                                      n                    ⁢                      ⅇ                          (                                                -                  E                                /                RT                            )                                                          (        3        )            
The parameters E, Z and n are called kinetic constants and may be used to model the dependence of a chemical reaction on time and temperature.
Thermogravimetry is used to obtain kinetic constants of decomposition or volatilization reactions using one of several common methods. One approach is known as the xe2x80x9cfactor jumpxe2x80x9d method where the temperature of the test specimen is xe2x80x9csteppedxe2x80x9d between two or more isothermally held temperatures in the weight loss region. The rate of weight loss (dxcex1/dt) at each of the isothermal regions may be substituted into equation (3), along with the respective isothermal temperature(T). Any two equations for adjacent steps may then be examined as their ratio and the resultant form may be solved for activation energy.                     E        =                                                            RT                1                            ⁢                              T                2                                                                    T                1                            -                              T                2                                              ⁡                      [                                          ln                ⁢                                                      ⅆ                                          α                      1                                                                            ⅆ                                          α                      2                                                                                  +                              ln                ⁢                                  xe2x80x83                                ⁢                                                      f                    ⁡                                          (                                              α                        2                                            )                                                                            f                    ⁡                                          (                                              α                        1                                            )                                                                                            ]                                              (        4        )            
Where:
dxcex11=rate of weight loss at temperature T1 
dxcex12=rate of weight loss at temperature T2 
f(xcex11)=kinetic expression at the value of dxcex11 
f(xcex12)=kinetic expression at the value of dxcex12 
Should the values for dxcex11 and dxcex12 be extrapolated to a common conversion level, then xcex11=xcex12 and ln [f(xcex11)/f(xcex12)]=0, reducing equation (4) to a more easily evaluated form:                     E        =                                            RT              1                        ⁢                          T              2                        ⁢                          ln              ⁡                              (                                                      ⅆ                                          α                      1                                                        /                                      ⅆ                                          α                      s                                                                      )                                                                        T              1                        -                          T              2                                                          (        5        )            
In thermal analysis, the temperature rate of change is a forcing function (or independent parameter) which produces some physical or chemical change in a test specimen resulting in a measured response (or dependent experimental) parameter such as weight change. A linear temperature ramp is the most commonly used of these forcing functions. U.S. Pat. No. 5,224,775, which is incorporated by reference herein, however, introduced to thermal analysis (including thermogravimetry), the concept of a modulated-temperature forcing function. In the modulated temperature approach, a linear temperature ramp is modulated with a sinusoidal heating rate oscillation. This periodic temperature function produces corresponding oscillatory output response signal proportional to some physical property of the material under test. Deconvolution of the resultant experimental parameter signals leads to analytical information unavailable from the linear ramp forcing function alone.
In this invention, a sinusoidal heating rate oscillation is applied to thermogravimetry to obtain dependent parameters signals useful for the obtaining of kinetic information. Specifically, if the temperature is changed in an sinusoidal fashion around an average temperature (T), then the values of the peak temperatures may be given (T+A) and (Txe2x88x92A), where A is the half peak-to-peak amplitude. This forcing function produces a corresponding oscillatory rate of weight change and logarithm of the rate of weight change response signals. These terms may be substituted into equation (4) to obtain:                     E        =                                            R              ⁡                              (                                                      T                    2                                    -                                      A                    2                                                  )                                                    2              ⁢              A                                ⁡                      [                                          ln                ⁢                                                      ⅆ                                          α                      1                                                                            ⅆ                                          α                      2                                                                                  +                              ln                ⁢                                  xe2x80x83                                ⁢                                                      f                    ⁡                                          (                                              α                        2                                            )                                                                            f                    ⁡                                          (                                              α                        1                                            )                                                                                            ]                                              (        6        )            
A mathematical deconvolution technique, such as a discrete fast Fourier transformation, may be applied to the forcing and response functions to obtain average and amplitude values on a continuous bases. If average temperature oscillation (T), temperature amplitude (2A), rate of weight loss (dxcex1/dt) and amplitude of the logarithm of rate of weight loss [L=ln (dxcex11/dxcex12)] are obtained for constant conversation values [i.e., xcex11=xcex12 and ln[f(xcex11)/f(xcex12)=0], equation (6) reduces to:                     E        =                                            R              ⁡                              (                                                      T                    2                                    -                                      A                    2                                                  )                                      ⁢            L                                2            ⁢            A                                              (        7        )            
This equation is independent of the form of the reaction equation and may be said to be model independent. If, however, a model is selected then other kinetic parameters, such as the pre-exponential factor and reaction order, may be obtained. Homogeneous decomposition or volatilization reactions are almost universally found to follow first order kinetics where n=1 and the logarithm of the pre-exponential factor equals ln [dxcex1/(1xe2x88x92xcex1)]+E/RT.
The use of continuously deconvoluted, or separated, values of the oscillatory forcing functions and corresponding oscillatory response provides, then, for the continuous generation of activation energy and pre-exponential factor throughout the reaction range.
Further, U.S. Pat. No. 5,165,792, which is incorporated by reference herein, describes how the heating rate (temperature forcing function) in thermal analysis may be adjusted according to the rate of change of a dependent parameter. In thermogravimetry, the temperature of the experiment is adjusted to maintain an average rate of weight change. A second part of this invention involves the control of the average experimental temperature using the average rate of weight change generated by the deconvolution.
According to equation (1), the rate of the reaction will decrease as the amount of reactant is consumed. To compensate for this rate of weight change reduction, the average temperature is adjusted during the reaction according to the average rate of weight change obtained from the deconvoluted response signal. This provides a smooth, continuous, and rapid change in the average temperature during regions where no weight change is observed but then a decrease in the average temperature change rate where weight changes reaction occur. Once a weight change is complete, the average temperature of the modulated temperature experiment is automatically increased until another weight change region is observed or the maximum temperature of the experiment is reached.